Journal of Uncertain Systems, Vol5 No1, 2011

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From this observation, Pawlak [15] defined lower approximation set X of X to be the set of ... The set of rough sets can be defined as s ={(X ,X ↑) : X ⊆ U}.
Journal of Uncertain Systems Vol.5, No.1, pp.72-80, 2011 Online at: www.jus.org.uk

Rough Set Approach on Lattice Dipankar Rana, Sankar Kumar Roy∗ Department of Applied Mathematics with Oceanology and Computer Programming Vidyasagar University, Midnapore-721102, West Bengal, India Received 21 July 2009; Revised 23 April 2010

Abstract This paper deals with rough set approach on lattice theory. We represent the lattices for rough sets determined by an equivalence relation. Without any loss of generality, we have defined the rough set as a pair of sets (lower approximation set, upper approximation set) and then we showed that the collection of all rough sets of an approximations by an equivalence relation form a lattice by some order relation. In this paper we are able to deal with information sources in a set-theoretic manner. We also given an integrated approach to form lattices by choice function and lattice structure in rough set theory. The simple notion of this paper is to show the lattice structure in rough set theory by using indiscernible equivalence relation. Some important results are also proved. Finally, some examples are considered to illustrate the paper. c

2011 World Academic Press, UK. All rights reserved. Keywords: rough set, choice function, equivalence class, indiscernibility relation, lattice

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Preliminaries

The lattice is one of the most widely discussed and studied structure in the classical algebraic theory, both as a specific algebra with a carrier and two binary operations, and as a relational structure-a specific ordered set [2]. The lattice as a poset will be denoted by (L, ≤), and the lattice as an algebra by (L, ∧, ∨). We write simply L to denote the lattice in both senses. A poset (L, ≤) is a lattice if sup{a, b} and inf {a, b} exist for all a, b ∈ L.

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Introduction

The original setting of the theory of rough sets was introduced by Z. Pawlak [15], assumed that sets are chosen from a universe U , but that elements of U can be specified only upto an indiscernibility equivalence relation E on U . If a subset X ⊆ U contains an element indiscernible from some elements not in X, then X is rough. Also a rough set X is described by two approximations. Basically, in rough set theory, it is assume that our knowledge is restricted by an indiscernibility relation. An indiscernibility relation is an equivalence relation E such that two elements of an universe of discourse U are E-equivalent if we can not distinguish these two elements by their properties known by us. By means of an indiscernibility relation E, we can partition the elements of U into three disjoint classes respect to any set X ⊆ U , defined as follows: • The elements which are certainly in X. These are elements x ∈ U whose E-class x/E is included in X. • The elements which certainly are not in X. These are elements x ∈ U such that their E-class x/E is included in X co , which is the complement of X. • The elements which are possibly belongs to X. These are elements whose E-class intersects with both X and X co . In other words, x/E is not included in X nor in X co . ∗ Corresponding

author. Email: [email protected] (S.K. Roy); Cell No: (+91)9434217733.

Journal of Uncertain Systems, Vol.5, No.1, pp.72-80, 2011

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From this observation, Pawlak [15] defined lower approximation set X ↓ of X to be the set of those elements x ∈ U whose E-class is included in X, i.e, X ↓= {x ∈ U : x/E ⊆ X} and for the upper approximation set X ↑ of X consists of elements x ∈ U whose E-class intersect with X, i.e, X ↑= {x ∈ U : x/E ∩ X 6= ∅}. The difference between X ↓ and X ↑ treated as the actual area of uncertainty. The study of lattices in rough set theory was initiated by Iwiniski [7]. He noticed that rough sets can be represented by their approximations. The set of rough sets can be defined as